// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_QUATERNION_H
#define EIGEN_QUATERNION_H
namespace Eigen {

/***************************************************************************
* Definition of QuaternionBase<Derived>
* The implementation is at the end of the file
***************************************************************************/

namespace internal {
    template <typename Other, int OtherRows = Other::RowsAtCompileTime, int OtherCols = Other::ColsAtCompileTime> struct quaternionbase_assign_impl;
}

/** \geometry_module \ingroup Geometry_Module
  * \class QuaternionBase
  * \brief Base class for quaternion expressions
  * \tparam Derived derived type (CRTP)
  * \sa class Quaternion
  */
template <class Derived> class QuaternionBase : public RotationBase<Derived, 3>
{
public:
    typedef RotationBase<Derived, 3> Base;

    using Base::operator*;
    using Base::derived;

    typedef typename internal::traits<Derived>::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef typename internal::traits<Derived>::Coefficients Coefficients;
    typedef typename Coefficients::CoeffReturnType CoeffReturnType;
    typedef typename internal::conditional<bool(internal::traits<Derived>::Flags& LvalueBit), Scalar&, CoeffReturnType>::type NonConstCoeffReturnType;

    enum
    {
        Flags = Eigen::internal::traits<Derived>::Flags
    };

    // typedef typename Matrix<Scalar,4,1> Coefficients;
    /** the type of a 3D vector */
    typedef Matrix<Scalar, 3, 1> Vector3;
    /** the equivalent rotation matrix type */
    typedef Matrix<Scalar, 3, 3> Matrix3;
    /** the equivalent angle-axis type */
    typedef AngleAxis<Scalar> AngleAxisType;

    /** \returns the \c x coefficient */
    EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); }
    /** \returns the \c y coefficient */
    EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); }
    /** \returns the \c z coefficient */
    EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); }
    /** \returns the \c w coefficient */
    EIGEN_DEVICE_FUNC inline CoeffReturnType w() const { return this->derived().coeffs().coeff(3); }

    /** \returns a reference to the \c x coefficient (if Derived is a non-const lvalue) */
    EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType x() { return this->derived().coeffs().x(); }
    /** \returns a reference to the \c y coefficient (if Derived is a non-const lvalue) */
    EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType y() { return this->derived().coeffs().y(); }
    /** \returns a reference to the \c z coefficient (if Derived is a non-const lvalue) */
    EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType z() { return this->derived().coeffs().z(); }
    /** \returns a reference to the \c w coefficient (if Derived is a non-const lvalue) */
    EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType w() { return this->derived().coeffs().w(); }

    /** \returns a read-only vector expression of the imaginary part (x,y,z) */
    EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients, 3> vec() const { return coeffs().template head<3>(); }

    /** \returns a vector expression of the imaginary part (x,y,z) */
    EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients, 3> vec() { return coeffs().template head<3>(); }

    /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
    EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }

    /** \returns a vector expression of the coefficients (x,y,z,w) */
    EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); }

    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other);
    template <class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator=(const QuaternionBase<OtherDerived>& other);

    // disabled this copy operator as it is giving very strange compilation errors when compiling
    // test_stdvector with GCC 4.4.2. This looks like a GCC bug though, so feel free to re-enable it if it's
    // useful; however notice that we already have the templated operator= above and e.g. in MatrixBase
    // we didn't have to add, in addition to templated operator=, such a non-templated copy operator.
    //  Derived& operator=(const QuaternionBase& other)
    //  { return operator=<Derived>(other); }

    EIGEN_DEVICE_FUNC Derived& operator=(const AngleAxisType& aa);
    template <class OtherDerived> EIGEN_DEVICE_FUNC Derived& operator=(const MatrixBase<OtherDerived>& m);

    /** \returns a quaternion representing an identity rotation
    * \sa MatrixBase::Identity()
    */
    EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }

    /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
    */
    EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity()
    {
        coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1);
        return *this;
    }

    /** \returns the squared norm of the quaternion's coefficients
    * \sa QuaternionBase::norm(), MatrixBase::squaredNorm()
    */
    EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }

    /** \returns the norm of the quaternion's coefficients
    * \sa QuaternionBase::squaredNorm(), MatrixBase::norm()
    */
    EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }

    /** Normalizes the quaternion \c *this
    * \sa normalized(), MatrixBase::normalize() */
    EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
    /** \returns a normalized copy of \c *this
    * \sa normalize(), MatrixBase::normalized() */
    EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }

    /** \returns the dot product of \c *this and \a other
    * Geometrically speaking, the dot product of two unit quaternions
    * corresponds to the cosine of half the angle between the two rotations.
    * \sa angularDistance()
    */
    template <class OtherDerived> EIGEN_DEVICE_FUNC inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); }

    template <class OtherDerived> EIGEN_DEVICE_FUNC Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const;

    /** \returns an equivalent 3x3 rotation matrix */
    EIGEN_DEVICE_FUNC inline Matrix3 toRotationMatrix() const;

    /** \returns the quaternion which transform \a a into \a b through a rotation */
    template <typename Derived1, typename Derived2> EIGEN_DEVICE_FUNC Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);

    template <class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<Scalar> operator*(const QuaternionBase<OtherDerived>& q) const;
    template <class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator*=(const QuaternionBase<OtherDerived>& q);

    /** \returns the quaternion describing the inverse rotation */
    EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const;

    /** \returns the conjugated quaternion */
    EIGEN_DEVICE_FUNC Quaternion<Scalar> conjugate() const;

    template <class OtherDerived> EIGEN_DEVICE_FUNC Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const;

    /** \returns true if each coefficients of \c *this and \a other are all exactly equal.
    * \warning When using floating point scalar values you probably should rather use a
    *          fuzzy comparison such as isApprox()
    * \sa isApprox(), operator!= */
    template <class OtherDerived> EIGEN_DEVICE_FUNC inline bool operator==(const QuaternionBase<OtherDerived>& other) const
    {
        return coeffs() == other.coeffs();
    }

    /** \returns true if at least one pair of coefficients of \c *this and \a other are not exactly equal to each other.
    * \warning When using floating point scalar values you probably should rather use a
    *          fuzzy comparison such as isApprox()
    * \sa isApprox(), operator== */
    template <class OtherDerived> EIGEN_DEVICE_FUNC inline bool operator!=(const QuaternionBase<OtherDerived>& other) const
    {
        return coeffs() != other.coeffs();
    }

    /** \returns \c true if \c *this is approximately equal to \a other, within the precision
    * determined by \a prec.
    *
    * \sa MatrixBase::isApprox() */
    template <class OtherDerived>
    EIGEN_DEVICE_FUNC bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const
    {
        return coeffs().isApprox(other.coeffs(), prec);
    }

    /** return the result vector of \a v through the rotation*/
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** \returns \c *this with scalar type casted to \a NewScalarType
    *
    * Note that if \a NewScalarType is equal to the current scalar type of \c *this
    * then this function smartly returns a const reference to \c *this.
    */
    template <typename NewScalarType> EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Derived, Quaternion<NewScalarType>>::type cast() const;

#else

    template <typename NewScalarType>
    EIGEN_DEVICE_FUNC inline typename internal::enable_if<internal::is_same<Scalar, NewScalarType>::value, const Derived&>::type cast() const
    {
        return derived();
    }

    template <typename NewScalarType>
    EIGEN_DEVICE_FUNC inline typename internal::enable_if<!internal::is_same<Scalar, NewScalarType>::value, Quaternion<NewScalarType>>::type cast() const
    {
        return Quaternion<NewScalarType>(coeffs().template cast<NewScalarType>());
    }
#endif

#ifndef EIGEN_NO_IO
    friend std::ostream& operator<<(std::ostream& s, const QuaternionBase<Derived>& q)
    {
        s << q.x() << "i + " << q.y() << "j + " << q.z() << "k"
          << " + " << q.w();
        return s;
    }
#endif

#ifdef EIGEN_QUATERNIONBASE_PLUGIN
#include EIGEN_QUATERNIONBASE_PLUGIN
#endif
protected:
    EIGEN_DEFAULT_COPY_CONSTRUCTOR(QuaternionBase)
    EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(QuaternionBase)
};

/***************************************************************************
* Definition/implementation of Quaternion<Scalar>
***************************************************************************/

/** \geometry_module \ingroup Geometry_Module
  *
  * \class Quaternion
  *
  * \brief The quaternion class used to represent 3D orientations and rotations
  *
  * \tparam _Scalar the scalar type, i.e., the type of the coefficients
  * \tparam _Options controls the memory alignment of the coefficients. Can be \# AutoAlign or \# DontAlign. Default is AutoAlign.
  *
  * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
  * orientations and rotations of objects in three dimensions. Compared to other representations
  * like Euler angles or 3x3 matrices, quaternions offer the following advantages:
  * \li \b compact storage (4 scalars)
  * \li \b efficient to compose (28 flops),
  * \li \b stable spherical interpolation
  *
  * The following two typedefs are provided for convenience:
  * \li \c Quaternionf for \c float
  * \li \c Quaterniond for \c double
  *
  * \warning Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized.
  *
  * \sa  class AngleAxis, class Transform
  */

namespace internal {
    template <typename _Scalar, int _Options> struct traits<Quaternion<_Scalar, _Options>>
    {
        typedef Quaternion<_Scalar, _Options> PlainObject;
        typedef _Scalar Scalar;
        typedef Matrix<_Scalar, 4, 1, _Options> Coefficients;
        enum
        {
            Alignment = internal::traits<Coefficients>::Alignment,
            Flags = LvalueBit
        };
    };
}  // namespace internal

template <typename _Scalar, int _Options> class Quaternion : public QuaternionBase<Quaternion<_Scalar, _Options>>
{
public:
    typedef QuaternionBase<Quaternion<_Scalar, _Options>> Base;
    enum
    {
        NeedsAlignment = internal::traits<Quaternion>::Alignment > 0
    };

    typedef _Scalar Scalar;

    EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion)
    using Base::operator*=;

    typedef typename internal::traits<Quaternion>::Coefficients Coefficients;
    typedef typename Base::AngleAxisType AngleAxisType;

    /** Default constructor leaving the quaternion uninitialized. */
    EIGEN_DEVICE_FUNC inline Quaternion() {}

    /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
    * its four coefficients \a w, \a x, \a y and \a z.
    *
    * \warning Note the order of the arguments: the real \a w coefficient first,
    * while internally the coefficients are stored in the following order:
    * [\c x, \c y, \c z, \c w]
    */
    EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w) {}

    /** Constructs and initialize a quaternion from the array data */
    EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {}

    /** Copy constructor */
    template <class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }

    /** Constructs and initializes a quaternion from the angle-axis \a aa */
    EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }

    /** Constructs and initializes a quaternion from either:
    *  - a rotation matrix expression,
    *  - a 4D vector expression representing quaternion coefficients.
    */
    template <typename Derived> EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }

    /** Explicit copy constructor with scalar conversion */
    template <typename OtherScalar, int OtherOptions> EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
    {
        m_coeffs = other.coeffs().template cast<Scalar>();
    }

#if EIGEN_HAS_RVALUE_REFERENCES
    // We define a copy constructor, which means we don't get an implicit move constructor or assignment operator.
    /** Default move constructor */
    EIGEN_DEVICE_FUNC inline Quaternion(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_constructible<Scalar>::value)
        : m_coeffs(std::move(other.coeffs()))
    {
    }

    /** Default move assignment operator */
    EIGEN_DEVICE_FUNC Quaternion& operator=(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_assignable<Scalar>::value)
    {
        m_coeffs = std::move(other.coeffs());
        return *this;
    }
#endif

    EIGEN_DEVICE_FUNC static Quaternion UnitRandom();

    template <typename Derived1, typename Derived2>
    EIGEN_DEVICE_FUNC static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);

    EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
    EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(NeedsAlignment))

#ifdef EIGEN_QUATERNION_PLUGIN
#include EIGEN_QUATERNION_PLUGIN
#endif

protected:
    Coefficients m_coeffs;

#ifndef EIGEN_PARSED_BY_DOXYGEN
    static EIGEN_STRONG_INLINE void _check_template_params() { EIGEN_STATIC_ASSERT((_Options & DontAlign) == _Options, INVALID_MATRIX_TEMPLATE_PARAMETERS) }
#endif
};

/** \ingroup Geometry_Module
  * single precision quaternion type */
typedef Quaternion<float> Quaternionf;
/** \ingroup Geometry_Module
  * double precision quaternion type */
typedef Quaternion<double> Quaterniond;

/***************************************************************************
* Specialization of Map<Quaternion<Scalar>>
***************************************************************************/

namespace internal {
    template <typename _Scalar, int _Options>
    struct traits<Map<Quaternion<_Scalar>, _Options>> : traits<Quaternion<_Scalar, (int(_Options) & Aligned) == Aligned ? AutoAlign : DontAlign>>
    {
        typedef Map<Matrix<_Scalar, 4, 1>, _Options> Coefficients;
    };
}  // namespace internal

namespace internal {
    template <typename _Scalar, int _Options>
    struct traits<Map<const Quaternion<_Scalar>, _Options>> : traits<Quaternion<_Scalar, (int(_Options) & Aligned) == Aligned ? AutoAlign : DontAlign>>
    {
        typedef Map<const Matrix<_Scalar, 4, 1>, _Options> Coefficients;
        typedef traits<Quaternion<_Scalar, (int(_Options) & Aligned) == Aligned ? AutoAlign : DontAlign>> TraitsBase;
        enum
        {
            Flags = TraitsBase::Flags & ~LvalueBit
        };
    };
}  // namespace internal

/** \ingroup Geometry_Module
  * \brief Quaternion expression mapping a constant memory buffer
  *
  * \tparam _Scalar the type of the Quaternion coefficients
  * \tparam _Options see class Map
  *
  * This is a specialization of class Map for Quaternion. This class allows to view
  * a 4 scalar memory buffer as an Eigen's Quaternion object.
  *
  * \sa class Map, class Quaternion, class QuaternionBase
  */
template <typename _Scalar, int _Options> class Map<const Quaternion<_Scalar>, _Options> : public QuaternionBase<Map<const Quaternion<_Scalar>, _Options>>
{
public:
    typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options>> Base;

    typedef _Scalar Scalar;
    typedef typename internal::traits<Map>::Coefficients Coefficients;
    EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
    using Base::operator*=;

    /** Constructs a Mapped Quaternion object from the pointer \a coeffs
      *
      * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
      * \code *coeffs == {x, y, z, w} \endcode
      *
      * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
    EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}

    EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

protected:
    const Coefficients m_coeffs;
};

/** \ingroup Geometry_Module
  * \brief Expression of a quaternion from a memory buffer
  *
  * \tparam _Scalar the type of the Quaternion coefficients
  * \tparam _Options see class Map
  *
  * This is a specialization of class Map for Quaternion. This class allows to view
  * a 4 scalar memory buffer as an Eigen's  Quaternion object.
  *
  * \sa class Map, class Quaternion, class QuaternionBase
  */
template <typename _Scalar, int _Options> class Map<Quaternion<_Scalar>, _Options> : public QuaternionBase<Map<Quaternion<_Scalar>, _Options>>
{
public:
    typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options>> Base;

    typedef _Scalar Scalar;
    typedef typename internal::traits<Map>::Coefficients Coefficients;
    EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
    using Base::operator*=;

    /** Constructs a Mapped Quaternion object from the pointer \a coeffs
      *
      * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
      * \code *coeffs == {x, y, z, w} \endcode
      *
      * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
    EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {}

    EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
    EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

protected:
    Coefficients m_coeffs;
};

/** \ingroup Geometry_Module
  * Map an unaligned array of single precision scalars as a quaternion */
typedef Map<Quaternion<float>, 0> QuaternionMapf;
/** \ingroup Geometry_Module
  * Map an unaligned array of double precision scalars as a quaternion */
typedef Map<Quaternion<double>, 0> QuaternionMapd;
/** \ingroup Geometry_Module
  * Map a 16-byte aligned array of single precision scalars as a quaternion */
typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf;
/** \ingroup Geometry_Module
  * Map a 16-byte aligned array of double precision scalars as a quaternion */
typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd;

/***************************************************************************
* Implementation of QuaternionBase methods
***************************************************************************/

// Generic Quaternion * Quaternion product
// This product can be specialized for a given architecture via the Arch template argument.
namespace internal {
    template <int Arch, class Derived1, class Derived2, typename Scalar> struct quat_product
    {
        EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b)
        {
            return Quaternion<Scalar>(a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
                                      a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
                                      a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
                                      a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x());
        }
    };
}  // namespace internal

/** \returns the concatenation of two rotations as a quaternion-quaternion product */
template <class Derived>
template <class OtherDerived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::
operator*(const QuaternionBase<OtherDerived>& other) const
{
    EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
                        YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
    return internal::quat_product<Architecture::Target, Derived, OtherDerived, typename internal::traits<Derived>::Scalar>::run(*this, other);
}

/** \sa operator*(Quaternion) */
template <class Derived>
template <class OtherDerived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*=(const QuaternionBase<OtherDerived>& other)
{
    derived() = derived() * other.derived();
    return derived();
}

/** Rotation of a vector by a quaternion.
  * \remarks If the quaternion is used to rotate several points (>1)
  * then it is much more efficient to first convert it to a 3x3 Matrix.
  * Comparison of the operation cost for n transformations:
  *   - Quaternion2:    30n
  *   - Via a Matrix3: 24 + 15n
  */
template <class Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3 QuaternionBase<Derived>::_transformVector(const Vector3& v) const
{
    // Note that this algorithm comes from the optimization by hand
    // of the conversion to a Matrix followed by a Matrix/Vector product.
    // It appears to be much faster than the common algorithm found
    // in the literature (30 versus 39 flops). It also requires two
    // Vector3 as temporaries.
    Vector3 uv = this->vec().cross(v);
    uv += uv;
    return v + this->w() * uv + this->vec().cross(uv);
}

template <class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
{
    coeffs() = other.coeffs();
    return derived();
}

template <class Derived>
template <class OtherDerived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other)
{
    coeffs() = other.coeffs();
    return derived();
}

/** Set \c *this from an angle-axis \a aa and returns a reference to \c *this
  */
template <class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
{
    EIGEN_USING_STD(cos)
    EIGEN_USING_STD(sin)
    Scalar ha = Scalar(0.5) * aa.angle();  // Scalar(0.5) to suppress precision loss warnings
    this->w() = cos(ha);
    this->vec() = sin(ha) * aa.axis();
    return derived();
}

/** Set \c *this from the expression \a xpr:
  *   - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
  *   - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
  *     and \a xpr is converted to a quaternion
  */

template <class Derived>
template <class MatrixDerived>
EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
{
    EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
                        YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
    internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
    return derived();
}

/** Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to
  * be normalized, otherwise the result is undefined.
  */
template <class Derived> EIGEN_DEVICE_FUNC inline typename QuaternionBase<Derived>::Matrix3 QuaternionBase<Derived>::toRotationMatrix(void) const
{
    // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
    // if not inlined then the cost of the return by value is huge ~ +35%,
    // however, not inlining this function is an order of magnitude slower, so
    // it has to be inlined, and so the return by value is not an issue
    Matrix3 res;

    const Scalar tx = Scalar(2) * this->x();
    const Scalar ty = Scalar(2) * this->y();
    const Scalar tz = Scalar(2) * this->z();
    const Scalar twx = tx * this->w();
    const Scalar twy = ty * this->w();
    const Scalar twz = tz * this->w();
    const Scalar txx = tx * this->x();
    const Scalar txy = ty * this->x();
    const Scalar txz = tz * this->x();
    const Scalar tyy = ty * this->y();
    const Scalar tyz = tz * this->y();
    const Scalar tzz = tz * this->z();

    res.coeffRef(0, 0) = Scalar(1) - (tyy + tzz);
    res.coeffRef(0, 1) = txy - twz;
    res.coeffRef(0, 2) = txz + twy;
    res.coeffRef(1, 0) = txy + twz;
    res.coeffRef(1, 1) = Scalar(1) - (txx + tzz);
    res.coeffRef(1, 2) = tyz - twx;
    res.coeffRef(2, 0) = txz - twy;
    res.coeffRef(2, 1) = tyz + twx;
    res.coeffRef(2, 2) = Scalar(1) - (txx + tyy);

    return res;
}

/** Sets \c *this to be a quaternion representing a rotation between
  * the two arbitrary vectors \a a and \a b. In other words, the built
  * rotation represent a rotation sending the line of direction \a a
  * to the line of direction \a b, both lines passing through the origin.
  *
  * \returns a reference to \c *this.
  *
  * Note that the two input vectors do \b not have to be normalized, and
  * do not need to have the same norm.
  */
template <class Derived>
template <typename Derived1, typename Derived2>
EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
{
    EIGEN_USING_STD(sqrt)
    Vector3 v0 = a.normalized();
    Vector3 v1 = b.normalized();
    Scalar c = v1.dot(v0);

    // if dot == -1, vectors are nearly opposites
    // => accurately compute the rotation axis by computing the
    //    intersection of the two planes. This is done by solving:
    //       x^T v0 = 0
    //       x^T v1 = 0
    //    under the constraint:
    //       ||x|| = 1
    //    which yields a singular value problem
    if (c < Scalar(-1) + NumTraits<Scalar>::dummy_precision())
    {
        c = numext::maxi(c, Scalar(-1));
        Matrix<Scalar, 2, 3> m;
        m << v0.transpose(), v1.transpose();
        JacobiSVD<Matrix<Scalar, 2, 3>> svd(m, ComputeFullV);
        Vector3 axis = svd.matrixV().col(2);

        Scalar w2 = (Scalar(1) + c) * Scalar(0.5);
        this->w() = sqrt(w2);
        this->vec() = axis * sqrt(Scalar(1) - w2);
        return derived();
    }
    Vector3 axis = v0.cross(v1);
    Scalar s = sqrt((Scalar(1) + c) * Scalar(2));
    Scalar invs = Scalar(1) / s;
    this->vec() = axis * invs;
    this->w() = s * Scalar(0.5);

    return derived();
}

/** \returns a random unit quaternion following a uniform distribution law on SO(3)
  *
  * \note The implementation is based on http://planning.cs.uiuc.edu/node198.html
  */
template <typename Scalar, int Options> EIGEN_DEVICE_FUNC Quaternion<Scalar, Options> Quaternion<Scalar, Options>::UnitRandom()
{
    EIGEN_USING_STD(sqrt)
    EIGEN_USING_STD(sin)
    EIGEN_USING_STD(cos)
    const Scalar u1 = internal::random<Scalar>(0, 1), u2 = internal::random<Scalar>(0, 2 * EIGEN_PI), u3 = internal::random<Scalar>(0, 2 * EIGEN_PI);
    const Scalar a = sqrt(Scalar(1) - u1), b = sqrt(u1);
    return Quaternion(a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3));
}

/** Returns a quaternion representing a rotation between
  * the two arbitrary vectors \a a and \a b. In other words, the built
  * rotation represent a rotation sending the line of direction \a a
  * to the line of direction \a b, both lines passing through the origin.
  *
  * \returns resulting quaternion
  *
  * Note that the two input vectors do \b not have to be normalized, and
  * do not need to have the same norm.
  */
template <typename Scalar, int Options>
template <typename Derived1, typename Derived2>
EIGEN_DEVICE_FUNC Quaternion<Scalar, Options> Quaternion<Scalar, Options>::FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
{
    Quaternion quat;
    quat.setFromTwoVectors(a, b);
    return quat;
}

/** \returns the multiplicative inverse of \c *this
  * Note that in most cases, i.e., if you simply want the opposite rotation,
  * and/or the quaternion is normalized, then it is enough to use the conjugate.
  *
  * \sa QuaternionBase::conjugate()
  */
template <class Derived> EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::inverse() const
{
    // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite()  ??
    Scalar n2 = this->squaredNorm();
    if (n2 > Scalar(0))
        return Quaternion<Scalar>(conjugate().coeffs() / n2);
    else
    {
        // return an invalid result to flag the error
        return Quaternion<Scalar>(Coefficients::Zero());
    }
}

// Generic conjugate of a Quaternion
namespace internal {
    template <int Arch, class Derived, typename Scalar> struct quat_conj
    {
        EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived>& q)
        {
            return Quaternion<Scalar>(q.w(), -q.x(), -q.y(), -q.z());
        }
    };
}  // namespace internal

/** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
  * if the quaternion is normalized.
  * The conjugate of a quaternion represents the opposite rotation.
  *
  * \sa Quaternion2::inverse()
  */
template <class Derived> EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::conjugate() const
{
    return internal::quat_conj<Architecture::Target, Derived, typename internal::traits<Derived>::Scalar>::run(*this);
}

/** \returns the angle (in radian) between two rotations
  * \sa dot()
  */
template <class Derived>
template <class OtherDerived>
EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Scalar QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const
{
    EIGEN_USING_STD(atan2)
    Quaternion<Scalar> d = (*this) * other.conjugate();
    return Scalar(2) * atan2(d.vec().norm(), numext::abs(d.w()));
}

/** \returns the spherical linear interpolation between the two quaternions
  * \c *this and \a other at the parameter \a t in [0;1].
  * 
  * This represents an interpolation for a constant motion between \c *this and \a other,
  * see also http://en.wikipedia.org/wiki/Slerp.
  */
template <class Derived>
template <class OtherDerived>
EIGEN_DEVICE_FUNC Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::slerp(const Scalar& t,
                                                                                                        const QuaternionBase<OtherDerived>& other) const
{
    EIGEN_USING_STD(acos)
    EIGEN_USING_STD(sin)
    const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
    Scalar d = this->dot(other);
    Scalar absD = numext::abs(d);

    Scalar scale0;
    Scalar scale1;

    if (absD >= one)
    {
        scale0 = Scalar(1) - t;
        scale1 = t;
    }
    else
    {
        // theta is the angle between the 2 quaternions
        Scalar theta = acos(absD);
        Scalar sinTheta = sin(theta);

        scale0 = sin((Scalar(1) - t) * theta) / sinTheta;
        scale1 = sin((t * theta)) / sinTheta;
    }
    if (d < Scalar(0))
        scale1 = -scale1;

    return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
}

namespace internal {

    // set from a rotation matrix
    template <typename Other> struct quaternionbase_assign_impl<Other, 3, 3>
    {
        typedef typename Other::Scalar Scalar;
        template <class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& a_mat)
        {
            const typename internal::nested_eval<Other, 2>::type mat(a_mat);
            EIGEN_USING_STD(sqrt)
            // This algorithm comes from  "Quaternion Calculus and Fast Animation",
            // Ken Shoemake, 1987 SIGGRAPH course notes
            Scalar t = mat.trace();
            if (t > Scalar(0))
            {
                t = sqrt(t + Scalar(1.0));
                q.w() = Scalar(0.5) * t;
                t = Scalar(0.5) / t;
                q.x() = (mat.coeff(2, 1) - mat.coeff(1, 2)) * t;
                q.y() = (mat.coeff(0, 2) - mat.coeff(2, 0)) * t;
                q.z() = (mat.coeff(1, 0) - mat.coeff(0, 1)) * t;
            }
            else
            {
                Index i = 0;
                if (mat.coeff(1, 1) > mat.coeff(0, 0))
                    i = 1;
                if (mat.coeff(2, 2) > mat.coeff(i, i))
                    i = 2;
                Index j = (i + 1) % 3;
                Index k = (j + 1) % 3;

                t = sqrt(mat.coeff(i, i) - mat.coeff(j, j) - mat.coeff(k, k) + Scalar(1.0));
                q.coeffs().coeffRef(i) = Scalar(0.5) * t;
                t = Scalar(0.5) / t;
                q.w() = (mat.coeff(k, j) - mat.coeff(j, k)) * t;
                q.coeffs().coeffRef(j) = (mat.coeff(j, i) + mat.coeff(i, j)) * t;
                q.coeffs().coeffRef(k) = (mat.coeff(k, i) + mat.coeff(i, k)) * t;
            }
        }
    };

    // set from a vector of coefficients assumed to be a quaternion
    template <typename Other> struct quaternionbase_assign_impl<Other, 4, 1>
    {
        typedef typename Other::Scalar Scalar;
        template <class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& vec) { q.coeffs() = vec; }
    };

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_QUATERNION_H
